A two-dimensional (2D) electron system can be used to study quasiparticles with various statistics. Three-dimensional electron complexes can be either fermionic, including electrons themselves, holes, trions, plasmarons, etc., or bosonic, involving excitons, plasmons, biexcitons, plexcitons, etc. On the contrary, a 2D electron system placed in an external magnetic field allows a fractional (anionic) statistics of quasiparticles intermediate between the bosonic and fermionic limits [1]. The possibility of experimental implementation of the anyonic statistics in two-dimensional electron systems was recognized soon after Laughlin’s pioneering work [2] describing the properties of electron systems for several fractional quantum Hall-effect (FQHE) states. However, it has been proven only recently that certain quasiparticles in two-dimensional electron systems are indeed anyons. The behavior of charged quasiparticles in a 1/3 FQHE state was found to differ from that with Bose–Einstein or Fermi statistics. Their behavior is instead described by π/3 statistics [3, 4]. Moreover, more complicated statistics including non-Abelian statistics are expected in FQHE states [5].

What are neutral excitations in the bulk of such anyonic matter. It is commonly accepted that magnetorotons, i.e., neutral excitations with a unit orbital angular momentum [6, 7], are bosons. However, experiments with magnetorotons are difficult because they are well-defined only at large 2D momenta about the inverse magnetic length. These excitations at zero momentum (most appropriate for an experimental study) enter the multiroton continuum and decay [8, 9]. At the same time, the theoretical analysis predicts a “magneto-graviton” branch, which is another low-energy boson branch of neutral excitations existing at zero momentum [10–12].

Magneto-gravitons are among the most exotic quasiparticles in solid-state physics. They are conventionally described in terms of the perturbations of the space metric introduced for a system of FQHE quasiparticles [10]. They have an orbital angular momentum of 2. If magneto-gravitons were detected experimentally, they would rapidly decay to the ground state [13]. However, it has been found that spin-1 analogs of magneto-gravitons (called spin-magneto-gravitons) can be the lowest-energy zero-momentum neutral excitations at certain parameters of the confining potential in the 2D system. In [14], we created the experimental conditions necessary to observe an ensemble of spin-magneto-gravitons. In this work, we study the optical properties of the excited Laughlin liquid with spin-magneto-gravitons and detect a new effect of a nonlinear optical response of the excited Laughlin liquid.

To study an ensemble of spin-magneto-gravitons, we used a heterostructure with an 18-nm-wide GaAs/AlGaAs quantum well with an electron density of 8.4 × 1010 cm−2. The sample was placed in a cryostat with pumped 3He vapors to maintain the sample at a temperature of 0. 5 K. The optical measurements were carried out in an external magnetic field up to 14 T in two different experimental configurations. In the first, two-fiber, configuration (I), one waveguide delivered the pump laser radiation to the sample, and the other was used to collect the reflected light and to guide it to the entrance slit of a spectrometer. In the second configuration (II), an optical window was used both to deliver the pump light to the sample and to collect the scattered light. The uniformly excited region in the first experimental configuration had a diameter of 600 μm, whereas the second experimental configuration allowed us to focus the pump laser beam into a much smaller spot less than 100 μm in diameter, though with a nonuniform spatial distribution. A spectroscopic resonant reflection technique was employed. The electron system was excited using a tunable cw laser source with a linewidth of 10 MHz. The contribution of the reflection from the sample surface was suppressed using crossed linear polarizers placed between the sample and exciting and collecting waveguides. We also measured the resonant photoluminescence (PL) and photoluminescence excitation (PLE) spectra of the electron system (see Fig. 1).

Fig. 1.
figure 1

(Color online) (a) (Blue circles) Photoluminescence excitation and (green line) resonant photoluminescence spectra of the Laughlin liquid measured at a temperature of 0. 5 K. Allowed 0e–0h optical transitions are marked as 0h↑ and 0h↓ depending on the electron spin in the conductance band of GaAs (the electron g-factor in GaAs is negative). The gray background marks the region of weakly allowed optical transitions from hole states with a predominantly nonzero orbital angular momentum in the valence band to the conduction band. (b) Resonant reflection (RR) spectra of (red line) the Laughlin liquid at 0. 5 K, and (black line) the electron system at 1. 4 K, when the Laughlin liquid is destroyed. The inset shows the scheme for measuring the resonant reflection signal.

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Lines corresponding to the allowed two-particle optical transitions from the valence band to the lowest Landau level in the conduction band were observed in the PLE spectrum of the examined quantum well. Hole states in the light- and heavy-hole subbands of the valence band in the magnetic field are superpositions of the states on the n = 0, 1, 2, and 3 Landau levels with spin projections of –3/2, –1/2, 1/2, and 3/2 on the magnetic field. The merging of the finite-width Landau levels results in a continuous region of weakly allowed optical transitions with energies above the energies of the allowed optical transitions from the lowest Landau level of heavy holes to the lowest Landau level in the conduction band (see Fig. 1). Consequently, the (0e–0h) transitions between the lowest Landau levels in the resonant photoluminescence spectra have large oscillator strengths. Since the Laughlin liquid occupies the lowest spin sublevel of the lowest electron Landau level, its resonant photoluminescence intensity is much lower than that of the upper spin sublevel [14]. For the same reason, the resonant reflection spectrum of the electron system demonstrates only a single pronounced line (I) with an intensity much higher than those of the other resonant reflection lines. The nature of the reflection channel I becomes clear from the corresponding photoluminescence spectrum. The process can be described as the absorption of a photon from the laser source with the production of an electron–hole pair consisting of a hole from the lowest Landau level of the heavy hole subband of the valence band and an electron from the empty upper spin sublevel of the lowest Landau level in the conduction band followed by the recombination of the photoexcited electron–hole pair.

In addition to the major line I, the resonant reflection spectrum of the Laughlin liquid has a fairly intense line II whose energy does not correspond to a maximum in the density of the valence band states (see Fig. 1) [14]. This line corresponds to spin-magneto-gravitons in the Laughlin liquid. The intensity of line II varies strongly when additional laser radiation with the energy exceeding that of the (0e–0h) optical transition generates an ensemble of ultra-long-lived spin-magneto-gravitons.

Spin-magneto-gravitons are produced in relaxation processes (see Fig. 2). An electron from the valence band of a quantum well is transferred to the empty upper spin sublevel of the lowest Landau level through a weakly allowed optical transition. A photoexcited hole from the valence band relaxes to the upper spin sublevel of the lowest heavy-hole Landau level due to strong spin–orbit coupling in the valence band of GaAs, changing the total spin of the electron system. Then, the valence band hole recombines with an electron from the Laughlin liquid, is transformed to a Fermi hole, and occupies the place of an electron on the lowest spin sublevel of the lowest Landau level in the conduction band. As a result, a photoexcited electron and the Fermi hole can form a spin-magneto-graviton if a fraction of the relaxation energy necessary to excite the spin-magneto-graviton is transferred to electrons in the Laughlin liquid in the process of relaxation of the photoexcited hole in the valence band (see Fig. 2). We found that the growth rate of line II per unit power of additional laser radiation is the highest when the energy of the laser radiation coincides with the energy of line II [14]. Therefore, spin-magneto-gravitons are produced most efficiently in this optical transition, whereas spin-magneto-gravitons are produced most efficiently when the entire relaxation energy in the valence band is transferred to the excitation of electrons in the Laughlin liquid. In this case, no extra acoustic phonon is needed to dissipate the excess energy (see Fig. 2). This process is the resonant Stokes Raman scattering producing spin-magneto-gravitons.

Fig. 2.
figure 2

(Color online) (a) Intensities of the resonant reflection line II energy at the powers from 0. 5 to 56 μW. The diagram shows the schemes of optical transitions with the production of spin-magneto-gravitons (smg). (b) (Left axis, black curve) Pair correlation function g(r) in the ground state of the Laughlin liquid at a filling factor of 1/3. (Right axis, red solid line) Difference between the pair correlation functions in the excited and ground states plotted for the zero-momentum spin-magneto-graviton. The variable r is given in units of the magnetic length. The numerical calculation was carried out for ten electrons using the method described in [14].

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However, Stokes Raman scattering itself cannot lead to the appearance of a line in the reflection spectrum. For line II to appear, the inverse anti-Stokes Raman scattering from the excited Laughlin liquid is required. In this case, the concept of the “Fermi hole” in the Laughlin liquid that could be filled with a valence electron in anti-Stokes Raman scattering is no longer applicable because an excitation does not exist. As shown in Fig. 2, positive and negative charges in the spin-magneto-graviton are “spread out” in space over at least five magnetic lengths. Nevertheless, an optical transition from the valence band to the conduction band with the annihilation of the spin-magneto-graviton is possible as a reverse process to Stokes Raman scattering. Thus, two scattering processes can contribute to line II: (i) Stokes Raman scattering producing the spin-magneto-graviton in the ground state of the Laughlin liquid followed by anti-Stokes Raman scattering annihilating this spin-magneto-graviton (Stokes–anti-Stokes Raman scattering) and (ii) anti-Stokes Raman scattering in the excited state of the Laughlin liquid annihilating the existing spin-magneto-graviton followed by Stokes Raman scattering with the subsequent production of the identical spin-magneto-graviton (anti-Stokes–Stokes Raman scattering). It is important that both processes must be coherent to contribute to resonant reflection with zero energy (momentum) transfer.

The reflection line II intensity is determined not only by the two aforementioned processes but also by the “idle” process of resonant elastic backscattering from the Laughlin liquid. Coherent Stokes–anti-Stokes Raman scattering cannot be distinguished from the background idle process. However, the inverse anti-Stokes–Stokes Raman scattering is expected to cause a nonlinear dependence of the resonant reflection signal on the pump power if a single laser source is used both to excite spin-magneto-gravitons and to measure light scattering from the existing spin-magneto-gravitons. Indeed, a superlinear (quadratic) enhancement of scattering in channel II is observed above a certain threshold of the laser pump power when the single laser source with the photon energy equal to the energy of line II is used (see Figs. 2–4).

Fig. 3.
figure 3

(Color online) (Left axis) Resonant reflection signal intensity versus the laser pump power obtained for the (red filled circles) I and (red empty circles) II scattering channels. The intensity of the reflected signal is normalized by the coefficient of proportionality α from Eq. (1). Black lines are linear functions. The red line is the square function of the pump power. (Right axis) Photoluminescence excitation intensity measured at the energy of line II.

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Fig. 4.
figure 4

(Color online) (Empty circles) Resonant reflection intensity in channel II versus the laser pump power measured in a small laser excitation spot as described in the main text. The intensity of the reflected signal is normalized by the coefficient of proportionality α from Eq. (1). The green, black, and red lines are square root, linear, and square functions of the pump power, respectively. The inset presents the scheme of (a) resonant Stokes Raman scattering, (b) coherent Anti-Stokes–Stokes Raman scattering, and (c) resonant elastic backscattering (idle process) that are induced by a photon with the energy of line II.

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The nonlinearity of the reflection signal intensity is determined by Stokes scattering and coherent anti-Stokes–Stokes scattering presented in insets (a) and (b) of Fig. 4, respectively. The optical process presented in inset (b) and the idle process shown in inset (c) can be described by the equation

$$I = \alpha P + \;\beta Pn,$$
(1)

where the first and the second terms are the contributions from processes (c) and (b), respectively; I is the reflected light intensity; P is the laser pump power; n is the total number of spin-magneto-gravitons in the Laughlin liquid; and the coefficient α is the probability of light scattering in process (c) and takes into account the capability of our experimental setup to excite the electron system and to collect the reflected light. Since the exact α value is fundamentally unknown, only the normalized quantity I/α has a physical meaning.

Calculating the average number of excitations created in process (a) under steady-state pumping, we arrive at the equation

$$\frac{{dn}}{{dt}} = \gamma P - \frac{n}{\tau } = 0,$$
(2)

where τ is the lifetime of spin-magneto-gravitons and γ is the probability of exciting a zero-momentum spin-magneto-graviton in process (a) taking into account all unknown parameters of our experimental setup. Thus, we obtain

$$\frac{I}{\alpha } = P + \frac{{\beta \gamma \tau }}{\alpha }{{P}^{2}} = P + \rho {{P}^{2}},$$
(3)

where ρ = βγτ/α, which yields the desired P2 dependence.

The resulting equations well describe the dependence of the scattering signal intensity on the pump power. At the initial stage, when the number of spin-magneto-gravitons in the electron system is small, idle process (c) dominates in the reflection spectrum. At the same time, the photons that excite the electron system are involved in process (a), which leads to the production of spin-magneto-gravitons in the Laughlin liquid. A further increase in the pump power activates scattering channel (b), which is determined by the number of spin-magneto-gravitons in the Laughlin liquid. As a result, a quadratic dependence of the reflection signal is observed (see Fig. 3).

To exclude possible alternative processes that can lead to a nonlinear response in the reflection spectrum, we consider the known mechanisms of nonlinearity in multiparticle systems. There are two well-studied effects stimulating nonlinear light scattering, which are most prominent in atomic condensates [15]. The Bose–Einstein statistics of excited states is responsible for the enhancement of light scattering. Namely, the scattering process with the creation of an excitation involves a quantum state already occupied with N similar excitations is enhanced due to the Bose factor N + 1. In this case, scattering nonlinearity can also be observed in light absorption. Hence, the dependence of the scattered light intensity on the pump power is expected to be more complex than quadratic. However, we did not observe the deviation from the quadratic dependence over a wide range of pump powers.

We independently measured the PLE intensity at the energy of line II as a function of the pump power. The dependence of the PLE intensity in high-mobility AlGaAs/GaAs quantum wells on the pump power is expected to be the same as the absorption intensity because the nonradiative decay time for photoexcited electrons is much longer than the recombination time [16]. The PLE intensity is found to be a linear function of the pump power (see Fig. 3). Therefore, the observed nonlinear optical response from the excited Laughlin liquid is not due to the Bose statistics of excitations. At the same time, it cannot be attributed to the macrofilling of a particular photon mode involved in the scattering process because the given range of cw laser pump power densities is too narrow to detect this effect [15].

Processes (a) and (b) should be saturated when the number of spin-magneto-gravitons reaches the maximum allowed value (full saturation of the Laughlin liquid with excitations); i.e., a further increase in the intensity of laser pump power should not lead to the superlinear increase in the intensity of the resonant reflection signal. Indeed, as the laser excitation spot is significantly reduced and the pump power density is correspondingly increased, the dependence of the resonant reflection signal changes from quadratic to sublinear (see Fig. 4).

To summarize, we have discovered a quadratic component in the dependence of the reflection signal from the excited Laughlin liquid at a filling factor of 1/3. The experimental data indicates that this component is due to coherent anti-Stokes–Stokes scattering induced by a quasi-equilibrium ensemble of spin-magneto-gravitons, which are spin-1 neutral excitations. A similar light scattering process should also be observed in other systems. However, the ratio of the signals from coherent anti-Stokes–Stokes scattering and elastic backscattering should be so small that it would be very difficult to distinguish the optical signal of the former process against the background of the latter. A unique opportunity for the direct observation of this interesting physical phenomenon appears due to the broken time-reversal symmetry in the Laughlin liquid, the ultra-long lifetimes of spin-magneto-gravitons, and resonance conditions associated with the mixing of Landau levels of the heavy- and light-hole bands in GaAs/AlGaAs quantum wells. This effect in the Laughlin liquid makes it possible to measure the exact energy of a zero-momentum spin-magneto-graviton, which is still determined up to the single-particle electron g-factor [14]. The determined energy 1. 6 meV of the zero-momentum spin-magneto-graviton is very close to a value of 1. 57 meV obtained in the computer simulation of the system of ten electrons. Furthermore, using coherent anti-Stokes–Stokes scattering of light, we have experimentally determined the density of the total saturation of the Laughlin liquid with spin-magneto-gravitons, which allows one to study the excited Laughlin liquid as a new quasi-equilibrium state of the electron matter.